Tight bounds on adjacency labels for monotone graph classes

A class of graphs admits an adjacency labeling scheme of size $f(n)$, if the vertices of any $n$-vertex graph $G$ in the class can be assigned binary strings (aka labels) of length $f(n)$ so that the adjacency between each pair of vertices in $G$ can be determined only from their labels. The Implicit Graph Conjecture (IGC) claimed that any graph class which is hereditary (i.e. closed under taking induced subgraphs) and factorial (i.e. containing $2^{\Theta(n \log n)}$ graphs on $n$ vertices) admits an adjacency labeling scheme of order optimal size $O(\log n)$. After thirty years open, the IGC was recently disproved [Hatami and Hatami, FOCS 2022]. In this work we show that the IGC does not hold even for monotone graph classes, i.e. classes closed under taking subgraphs. More specifically, we show that there are monotone factorial graph classes for which the size of any adjacency labeling scheme is $\Omega(\log^2 n)$. Moreover, this is best possible, as any monotone factorial class admits an adjacency labeling scheme of size $O(\log^2 n)$. This is a consequence of our general result that establishes tight bounds on the size of adjacency labeling schemes for monotone graph classes: for any function $f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ with $\log x \leq f(x) \leq x^{1-\delta}$ for some constant $\delta > 0$, that satisfies some natural conditions, there exist monotone graph classes, in which the number of $n$-vertex graphs grows as $2^{O(nf(n))}$ and that do not admit adjacency labels of size at most $f(n) \log n$. On the other hand any such class admits adjacency labels of size $O(f(n)\log n)$, which is a factor of $\log n$ away from the order optimal bound $O(f(n))$. This is the first example of tight bounds on adjacency labels for graph classes that do not admit order optimal adjacency labeling schemes.